Question: $ F = \left[\begin{array}{rr}-1 & -2 \\ 2 & 2 \\ 1 & 1\end{array}\right]$ $ A = \left[\begin{array}{rr}3 & 4 \\ 5 & 0\end{array}\right]$ What is $ F A$ ?
Because $ F$ has dimensions $(3\times2)$ and $ A$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ F A = \left[\begin{array}{rr}{-1} & {-2} \\ {2} & {2} \\ \color{gray}{1} & \color{gray}{1}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{4} \\ {5} & \color{#DF0030}{0}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{3}+{-2}\cdot{5} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{3}+{-2}\cdot{5} & ? \\ {2}\cdot{3}+{2}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{3}+{-2}\cdot{5} & {-1}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{0} \\ {2}\cdot{3}+{2}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{3}+{-2}\cdot{5} & {-1}\cdot\color{#DF0030}{4}+{-2}\cdot\color{#DF0030}{0} \\ {2}\cdot{3}+{2}\cdot{5} & {2}\cdot\color{#DF0030}{4}+{2}\cdot\color{#DF0030}{0} \\ \color{gray}{1}\cdot{3}+\color{gray}{1}\cdot{5} & \color{gray}{1}\cdot\color{#DF0030}{4}+\color{gray}{1}\cdot\color{#DF0030}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-13 & -4 \\ 16 & 8 \\ 8 & 4\end{array}\right] $